Question

how can i prove this one

Answer 1

Okay, this is my proof. I'm not exactly sure if this is a viable proof, but I think it works.

$f(x) = lnx$
$f'(x) = \frac{1}{x}$

Hence, from x > 0, it is always increasing (gradient > 0)
y = lnx crosses the x-axis only once, so there is only one root.

Since x cannot be less than zero, as well as a monotonic increasing function for x > 0, and the fact that it crosses the x-axis once, then as x approaches 0 from the positive side, f(x) has to be approaching negative infinity.